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Greetings again. Here we compare steel to a titanium/tantalum alloy, Ti/Ta 70/30 Beta, which is used in surgical equipment and prosthetics. It interests me, not so much to propose it as tongue material, but because the tantalum increases the density well above the neat material. The modulus is only a third of that for steel, but its density is a little above that for steel, resulting in a modulus-to-density ratio about 0.3 that of steel. We can compare these results to those comparing Titanium to steel and find that the higher harmonic amplitudes in comparison to steel are very much less in the Ti/Ta alloy. The behavior with increasing bellows pressure shows the harmonic amplitudes making up the difference – just what happens in the Titanium/steel comparison; however, since the higher harmonics of the alloy start with such low values, they don’t catch up to those of steel nearly as quickly or as completely as in the case of the neat material. Comparing these plots with the plots for brass/steel, we see an enormous difference. If we can discern an audible difference in timbre between brass and steel, according to these plots, and if our basic assumptions are correct, there should be an enormous difference in timbre between either Ti or its Ti/Ta alloy and either brass or steel. Another interesting outcome in the case of the alloy is that I couldn’t get a convergence with a solution for bellows pressure less than about 2.2 inch w.c. On the other hand, solutions continue further on for the higher bellows pressure than they do for steel and other materials. This might indicate a material that may not be able to play at very low bellows pressures, but may perform better than other materials at the highest bellows pressures. The last plot in these calculations show that the trend of increasing harmonic amplitude with bellows pressure continues for even the highest bellows pressures, for those above solutions for steel become impossible. We can make such plots for most any feasible tongue material, though at some point, the usefulness tapers off, at least if we are in search of a practical tongue material. The theoretical interest continues, as a way of getting a good intuitive feel for the effect of material properties on musical tone. At this point, I think we need wait for experimentalists to compare information they have from work already accomplished or from new work. There might be other materials I'd like to do calculations on, such as glass, and other geometric factors, such as tongue thickness and plate thickness. A docx file showing these plots are at: https://app.box.com/folder/79305691686 Best regards, Tom
Greetings Free Reed enthusiasts. I posted here about a year ago explaining an analysis I was undertaking on how the Western free reed works, starting from fundamental principles in Fluid Mechanics. I’m happy to announce that I’ve made progress to the point where resulting calculations reveal much about the physical working of this sound source, and I’d like to present some results here. Very briefly, the analysis starts with the Euler-Bernoulli wave equation for the vibration of the tongue and combines it with a physical model that describes the external forces acting on the tongue. These forces include the bellows pressure force, the viscous friction acting on the skin of the tongue produced by relative air flow, the aerodynamic “form drag” acting on the tongue, and the effect of the mean airflow through the slot. The model does not include inertial forces caused by air on the tongue, though it can be shown that these forces are small, especially if the vibration frequency is not too large. There are many different characteristics of this vibration, far too many to present here, but without getting into too much background or detail, I can present some interesting results. Many concertina players say they can hear a difference between the sound of brass tongues and steel tongues, and many of us have wondered how different materials would sound. In the case of Brass vs Steel, I uploaded a .docx file that viewers can view on a cloud storage site, with details below. This file contains harmonic amplitudes plotted on a log scale, in the same way that a Fourier spectrum is viewed. The amplitudes are normalized with respect to the fundamental. In order to understand what these harmonics are, consider first that the vibrating tongue can vibrate in different bending modes, each with its own frequency. All free reed vibration in musical instruments involve virtually only the first bending mode, in which the tongue bends in a smoothly curved fashion, with only one node, where vibration is zero, at the rivet. The fundamental frequency of the musical tone we hear is the same as the frequency of the first bending mode, also called a fundamental. These details are sometimes confused. The second bending mode, which can occur under spurious conditions, has a node at the rivet and a node roughly a third of the length away from the free end, and the tongue bends in snakelike fashion. Unlike a string, whose modes of vibration are easily excited and are spaced at frequencies that are closely integer multiples of the fundamental, the second mode of the tongue vibration is roughly six times the frequency of the first mode, with succeeding modes separated by growing frequency gaps. Now, within a given mode of vibration, the tongue can also vibrate with frequencies that are multiple integers – harmonics - of the bending mode frequency. Let’s focus on a tongue vibrating in its first bending mode, the usual case. Let’s also consider an isolated tongue, without the slot, vibrating as a simple cantilever. If the excitation is “gentle” enough, or when any initial “complicated” vibrations die out, the motion will be very close to that of a sine wave. This is why tuning forks sound so pure. However, when the tongue is placed over the slot and pushed by a bellows pressure, the excitation is not so gentle. The external forces on the tongue during its vibration change suddenly, depending where the tongue is in its cycle. These sudden, changing forces induce harmonics in the general motion, although compared to the fundamental motion, they are usually small. You can imagine, instead of the relatively gentle sinusoidal swinging, sudden minute lurches and hindrances superimposed on the motion. Although these adjustments are relatively small, they can be measured, and who can really say how much affect they have on air vibration, and thus on the sound we hear? And if you think about it, how else could brass make a different musical tone than steel, if not by the way it vibrates? We expect that somewhere in the frequency spectrum of the tongue motion, there are clues to why brass and steel sound differently. The plots in the .docx file compare the harmonic amplitudes of tongue motion between the usual ASTME 1095 spring steel and Brass 260, characterized only by their Young’s Modulus and density. Each plot is for a different bellows pressure, indicated on the plot. The log scale means there’s a factor of ten in displacement amplitude separating each integer on the vertical scale. For low bellows pressure, the first harmonic greatly dominates, indicating relatively pure sinusoidal motion. The 2.5 orders of magnitude on the plot is a factor of about 316 in amplitude between the fundamental and first overtone. Also, differences between steel and brass are unnoticeable. As bellows pressures increase, higher harmonics play larger roles, and the difference between brass and steel becomes evident. With minor exception, steel often dominates brass in the higher harmonics, especially for the lowest of those harmonics where differences occur, around the 4th or 5th. However, the first three harmonics or so never really show much difference. Because of the relative amplitudes, we are more likely to hear the differences in the 4th to 6th harmonics more than those in the higher harmonics. From these plots we see a demonstration of the “brighter” sound of steel, or the “mellower” sound of brass, with such differences becoming more pronounced at larger bellows pressure. But why does that occur? The answer to that lies in the details of the physical model, to be uncovered by further investigation. In short, it’s due to the nonlinear ways in which energy dissipation acts in the system. The URL for the .docx file is below, and any visitor should be able to view it. I’m not releasing it for download because I intend to publish these results in the future and I’d rather wait for public release. https://app.box.com/s/zchj1y5d4l65fttok3n4sofupzx2gsio Best regards, Tom www.bluesbox.biz
Greetings again to free reed enthusiasts. In this post, I’d like to show results comparing steel to carbon fiber. In thinking about this issue yesterday, I did a web search and found that people are now selling carbon fiber material in thin sheets – for very reasonable prices. This opens the possibility of experimenting with this material as tongue material. Though, I'm sure some makers already know this. I say this because of the very interesting results from the free reed physics model I’m working on. I put up another .docx file showing the difference between steel and carbon fiber vibration spectrum in the same way I showed for steel and brass, and the link is below. The results indicate that the primary material property that affects tongue vibration is the ratio of Young’s modulus to density. In 2012, I posted a survey of materials that one might consider making tongues from, based on the hypothesis that E/rho is the only material property you need to know as a measure of what the musical tone would be. The link is: https://www.concertina.net/forums/index.php?/topic/14568-reed-tongue-materials-a-survey/&tab=comments#comment-138942 and the original table is still available. That hypothesis is valid rigorously only for the free vibration of the tongue, which occurs after the transients have died out when you start vibration by plucking. I didn’t know what effect it would have in forced vibration; i.e., excitation by a bellows pressure. The physical model I now developed shows that there’s an influence by both E and rho separately, apart from their appearance in the ratio. But now with this complete theory we can calculate the effects of E and rho, along with all the other important parameters, and these calculations give support for the simple idea that the ratio is the primary influence. If we normalize the ratio E/rho for different materials using that for steel (divide all ratios by that for steel), we get 1 for steel, 0.497 for brass, and 6.8 for carbon fiber. There’s considerable variation for carbon, but I think this is a representative value. I’m assuming here that steel sounds brighter than brass, as reported. I assume further that the reason is because steel has about twice the ratio E/rho that brass has. This is true IF the character of these harmonics in tongue motion carry through to the musical tone, and the difference in harmonic amplitude is now firmly established. From the plots, we see that carbon, with an E/rho ratio over six times that of steel, produces higher harmonics that greatly dominate those for steel, even at relatively low bellows pressure. In FREE REED PHYSICS – 1, plots show that higher harmonics for steel dominate those for brass at higher bellows pressure. But with carbon vs steel, the dominance is much more, with carbon favored. As the reasoning goes, we thus expect that carbon would sound much brighter than steel, even much brighter than steel sounds in relation to brass. Of course, I could be wrong, and these tongue vibration harmonics don’t translate to musical tone. I’d be surprised because I don’t see any other way that tongue material could affect musical tone. The fact that players report clear differences in the sound of brass vs that of steel strongly indicates to me that plots such as these can lead to an educated guess on what different tongue materials sound like, just from knowing these two key properties. I did simple calculations on the tongue geometry required for a carbon tongue material. Using available thicknesses (0.5 mm and 1 mm), the lowest concertina pitches would require lengths around 4 inches at 0.5 mm thickness for 100 Hz. These lengths are probably too large, and thinner sheets would be required for shorter lengths. Perhaps the 0.5 mm size could be sanded down. Carbon is a material very easy to work with. The high end is more accommodating, requiring lengths around 5/8 inch at 1 mm thickness for 8700 Hz. A 1 mm thickness is also much thicker than existing steel construction, and that might introduce interesting issues with such short lengths, perhaps in connection with the plate thickness. We can of course now use the model for calculations involving different plate thickness. Another alternative is to make tongues at the extreme pitch ends out of steel. Of course, we can wonder just how bright a carbon fiber tongue could sound, and I encourage makers to give it a try, if they haven’t already. The docx file for Carbon and Steel is at: https://app.box.com/s/zchj1y5d4l65fttok3n4sofupzx2gsio Best regards, Tom